Interferometers are widely used to make precision measurements of displacement. FIG. 1 is a schematic drawing showing an example of a conventional high-stability plane mirror interferometer 10. Interferometer 10 is composed of a polarizing beam splitter (PBS) 20 having a single beam-splitting surface 22, a retroreflector 30, a fixed plane mirror 40, a movable plane mirror 50, a quarter-wave plate 44 and a quarter-wave plate 54. Fixed mirror 40 is in a fixed location relative to PBS 20 and has a plane reflective surface 42. Movable mirror 50 is affixed to a body (not shown) whose displacement is to be measured using interferometer 10 and has a plane reflective surface 52. The reflective surface 52 of movable mirror 50 is oriented orthogonally to the reflective surface 42 of fixed mirror 40 and to the direction of motion 56 of movable mirror 50.
PBS 20 is located such that the center of beam-splitting surface 22 is located at a point where a normal to the reflective surface 42 of fixed mirror 40 intersects with a normal to the reflective surface 52 of movable mirror 50. In the example shown, PBS 20 is oriented such that beam-splitting surface 22 is oriented at 450 with respect to the reflective surface 42 of fixed mirror 40. Other orientations of beam-splitting surface 22 are possible. Beam-splitting surface 22 reflects s-polarized light and transmits p-polarized light.
Quarter-wave plate 44 is interposed between PBS 20 and movable mirror 40. Quarter-wave plate 54 is interposed between PBS 20 and movable mirror 50. The surfaces of each quarter-wave plate are oriented parallel to the reflective surface of the respective mirror. Retroreflector 30 is located on the other side of PBS 20 from fixed mirror 40 and faces the fixed mirror. Retroreflector 30 is shown throughout this disclosure as a two-dimensional retroreflector, i.e., a prism, to simplify the drawings. More, typically, retroreflector 30 is a three-dimensional retroreflector, such as a corner cube.
FIG. 1 shows PBS 20, fixed mirror 40 and quarter-wave plates 44 and 54 as separate components to enable these components to be shown clearly. In embodiments in which PBS 20 is configured as a cube having a plane cube surface 24 and a plane cube surface 25 oriented at −45° and +45°, respectively, relative to beam-splitting surface 22, as exemplified in FIG. 1, quarter-wave plates 44 and 54 are affixed to plane cube surfaces 24 and 25, respectively. Additionally, fixed mirror 40 is typically embodied as a reflective layer deposited on the surface of quarter-wave plate 44 remote from cube surface 24.
FIG. 1 additionally shows retroreflector 30 juxtaposed with a cube surface 26 of PBS 20. Cube surface 26 is opposite cube surface 24. Retroreflector 30 may alternatively be spaced from cube surface 26.
A light source 60 is arranged to illuminate the beam-splitting surface 22 of PBS 20 with an incident beam 62 of light whose direction is orthogonal to the reflective surface 52 of movable mirror 50. Incident light beam 62 is composed of a first component having a first polarization and a first frequency and a second component having a second polarization and a second frequency. The first and second frequencies typically differ by a difference that typically ranges from a few MHz to a few tens of MHz. Light source 60 typically comprises a laser (not shown), typically a helium-neon laser.
A measurement beam 72 and a reference beam 74 derived by interferometer 10 from incident light beam 62 emerge from PBS 20 and illuminate a light sensor 70. Sensor 70 generates an electrical signal in response to the measurement beam and the reference beam.
Incident beam 62 is emitted by light source 60 at a location A and is incident at a location B on the beam-splitting surface 22 of PBS 20. The s-polarized component of incident beam 62 is reflected by beam-splitting surface 22 at location B and constitutes a reference beam 72. Reference beam 72 passes through quarter-wave plate 44 and is incident at a location C on the reflective surface 42 of fixed mirror 40.
After the first reflection by fixed mirror 40, reference beam 72 returns along a reciprocal path towards PBS 20. Reference beam 72 passes through quarter-wave plate 44 and is incident at a location D on beam-splitting surface 22. Location D is coincident with location B, described above. Letters indicating nominally-coincident locations are separated by commas in the Figures. Two passes through the quarter-wave plate change reference beam 72 from s-polarized to p-polarized. Consequently, beam-splitting surface 22 transmits the reference beam towards retroreflector 30.
Retroreflector 30 reflects reference beam 72 at locations E and F, after which the reference beam returns to PBS 20 along a path that is parallel to, but offset from, the path of the reference beam between locations D and E. Reference beam 72 is incident on the beam-splitting surface 22 of PBS 20, this time at a location G. Beam-splitting surface 22 transmits reference beam 72 towards fixed mirror 40. Reference beam 72 passes once more through quarter-wave plate 44, and is incident on the reflective surface 42 of fixed mirror 40 at a location H.
After the second reflection by fixed mirror 40, reference beam 72 returns along a reciprocal path towards PBS 20. Reference beam 72 passes through quarter-wave plate 44 a final time and is incident at a location I on the beam-splitting surface 22 of PBS 20. Location I is coincident with location G, described above. The final pass through quarter-wave plate 44 restores reference beam 72 to s-polarized. Consequently, beam-splitting surface 22 reflects reference beam 72, and the reference beam emerges from PBS 20 and is incident on sensor 70 at a location J.
The p-polarized component of incident beam 62 is also incident at location B on the beam-splitting surface 22 of PBS 20. The beam-splitting surface transmits the p-polarized component of incident beam 62 towards movable mirror 50 as a measurement beam 74. Measurement beam 74 passes through quarter-wave plate 54 and is incident at a location K on the reflective surface 52 of movable mirror 50.
After the first reflection by movable mirror 50, measurement beam 74 returns along a reciprocal path towards PBS 20. Measurement beam 74 passes again through quarter-wave plate 54 and is incident at a location L on beam-splitting surface 22. Location L is coincident with locations B and D, described above. Two passes through the quarter-wave plate 54 change measurement beam 74 from p-polarized to s-polarized. Consequently, beam-splitting surface 22 reflects measurement beam 74 towards retroreflector 30.
Retroreflector 30 reflects measurement beam 74 at locations M and N, which are coincident with locations E and F, respectively. Measurement beam 74 then returns to PBS 20, where it is once more incident on the beam-splitting surface 22 of PBS 20, this time at a location O. Location is coincident with locations G and I, described above. Beam-splitting surface 22 reflects measurement beam 74 towards movable mirror 50. Measurement beam 74 passes through quarter-wave plate 54 a third time, and is incident on the reflective surface 52 of movable mirror 50 at a location P.
After the second reflection by movable mirror 50, measurement beam 74 returns along a reciprocal path towards PBS 20. Measurement beam 74 passes a fourth time through the quarter-wave plate 54 and is incident at a location Q on the beam-splitting surface 22 of PBS 20. Location Q is coincident with locations G, I and O, described above. The final pass through quarter-wave plate 54 restores measurement beam 74 to p-polarized. Consequently, beam-splitting surface 22 transmits measurement beam 74, and the measurement beam emerges from PBS 20 and is incident on sensor 70 at a location R.
Location R is coincident with location J. Consequently, measurement beam 74 is nominally superposed with reference beam 72, and the beam axes of reference beam 72 and measurement beam 74 are parallel and coincident. Note that if fixed mirror 40 and movable mirror 50 are not ideally aligned, i.e., the mirrors are not perfectly orthogonal, the beam axes remain parallel, but one beam axis is spatially offset from the other.
Sensor 70 electrically detects the superposed light beams 72 and 74 to generate an electrical signal that includes a signal component generated by the sensor in response to interference between light beams 72, 74. This electrical signal component will be referred throughout this disclosure as a desired interference component. The amplitude of the desired interference component depends in part on two main factors, namely, the angle between the beam axes of light beams 72 and 74 and the distance between the beam axes of the light beams. The amplitude of the desired interference component is maximized when the beam axes of light beams 72 and 74 are parallel. The amplitude of the desired interference component generated in response to interference between two light beams, such as reference beam 72 and measurement beam 74, having a Gaussian intensity distribution falls off approximately as exp(−(2w/d)2), where d is the diameter of the beams, and w is the spatial separation between the beam axes of the beams, i.e., the spatial separation between the beam axis of reference beam 72 and the beam axis of measurement beam 74. In interferometer 10, spatial separation w is approximately 4Lθ, where L is approximately equal to the distance from the apex of retroreflector 30 to fixed mirror 40, and θ is the angle between fixed mirror 40 and an ideally-aligned fixed mirror 40, i.e., a fixed mirror 40 that is oriented exactly orthogonally with respect to movable mirror 50.
Imperfections in one of more of the components of a conventional interferometer, such as interferometer 10 shown in FIG. 1, typically subject the interferometer to ghost beams. FIGS. 2A and 2B are schematic drawings showing examples of ghost beams to which conventional interferometer 10 is typically subject. Examples of imperfections in one or more of the components of the interferometer are reflections at the cube surfaces 24, 26 of PBS 20, reflections at the surfaces of quarter-wave plates 44 and 54, transmittal of s-polarized light and reflection of p-polarized light at beam-splitting surface 22, and retroreflector 30 having imperfect polarization properties. FIG. 2A shows a ghost beam that originates as the result of reference beam 72 incident on beam-splitting surface 22 comprising an unwanted s-polarized component. This ghost beam will be referred to a ghost reference beam 82. FIG. 2B shows a ghost beam that originates as a result of measurement beam 74 comprising an unwanted p-polarized component. This ghost beam will be referred to a ghost measurement beam 84. Other ghost beams are possible.
FIG. 2A shows the path of ghost reference beam 82. Ghost reference beam 82 is a result of reference beam 72 incident at location G on the beam-splitting surface 22 of PBS 20 comprising an unwanted s-polarized component, typically due to retroreflector 30 having non-ideal polarization properties. Beam-splitting surface 22 reflects the s-polarized component as ghost reference beam 82.
Beam-splitting surface 22 reflects ghost reference beam 82 towards movable mirror 50. Ghost reference beam 82 is incident at a location S on the reflective surface 52 of movable mirror 50. Location S is coincident with location P, described above.
After reflection once by fixed mirror 40 and once by movable mirror 50, ghost reference beam 82 returns along a reciprocal path towards PBS 20. Ghost reference beam 82 passes a second time through quarter-wave plate 52 and is incident at a location T on the beam-splitting surface 22 of PBS 20. Location T is coincident with locations G, I, O and Q, described above. Passing twice through quarter-wave plate 54 twice changes ghost reference beam 82 to p-polarized. Consequently, beam-splitting surface 22 transmits ghost reference beam 82, and the ghost reference beam emerges from PBS 20 superposed with the desired beams, i.e., reference beam 72 and measurement beam 74. Ghost reference beam 82 is incident on sensor 70 at a location U, which is coincident with locations J and R. The beam axis of ghost reference beam 82 is parallel to and coincident with the beam axes of desired beams 72, 74. Consequently, ghost reference beam 82 interferes strongly with desired beams 72, 74 at the sensor.
Between light source 60 and sensor 70, ghost reference beam 82 is reflected once by fixed mirror 40 and once by movable mirror 50, whereas reference beam 72 is reflected twice by fixed mirror 40 and measurement beam 74 is reflected twice by movable mirror 50. Consequently, a given displacement of movable mirror 50 subjects ghost reference beam 82 to a phase change equal to one half of the phase change to which the same displacement of the movable mirror subjects measurement beam 74. Interference between ghost reference beam 82 and the desired beams 72, 74 imposes a cyclic error term with a period of λ/2, where λ is the nominal wavelength of the light output by light source 60, on the electrical signal generated by sensor 70. In other words, when movable mirror 50 is displaced by a distance equal to λ/2, ghost reference beam 82 imposes a full cycle of an approximately sinusoidal error on the electrical signal generated by sensor 70.
FIG. 2B shows the path of ghost measurement beam 84. Ghost measurement beam 84 is a result of measurement beam 74 incident at location O on beam-splitting surface 22 comprising an unwanted p-polarized component, typically due to retroreflector 30 having non-ideal polarization properties. Beam-splitting surface 22 transmits the p-polarized component of the measurement beam towards fixed mirror 40 as a ghost measurement beam 84.
Beam-splitting surface 22 transmits ghost measurement beam 84 towards fixed mirror 40. Ghost measurement beam 84 passes through quarter-wave plate 44 and is incident at a location V on the reflective surface 42 of fixed mirror 40. Location V is coincident with location H, described above.
After reflection once by movable mirror 50 and once by fixed mirror 40, ghost measurement beam 84 returns along a reciprocal path towards PBS 20. Ghost measurement beam passes a second time through quarter-wave plate 44 and is incident at a location W on the beam-splitting surface 22 of PBS 20. Location W is coincident with locations G, I, O and Q, described above. Passing through quarter-wave plate 44 twice changes ghost measurement beam 84 to s-polarized. Consequently, beam-splitting surface 22 reflects ghost measurement beam 84, and the ghost measurement beam emerges from PBS 20 superposed with the desired beams, i.e., reference beam 72 and measurement beam 74. Ghost measurement beam 84 is incident on sensor 70 at a location X, which is coincident with locations J and R. The beam axis of ghost measurement beam 84 is parallel to and coincident with the beam axes of desired beams 72, 74. Consequently, ghost beam 84 interferes strongly with desired beams 72, 74 at the sensor.
Between light source 60 and sensor 70, ghost measurement beam 84 is reflected once by fixed mirror 40 and once by movable mirror 50, whereas reference beam 72 is reflected twice by fixed mirror 40 and measurement beam 74 is reflected twice by movable mirror 50. Consequently, a given displacement of movable mirror 50 subjects ghost measurement beam to a phase change equal to one half of the phase change to which the same displacement of the movable mirror subjects measurement beam 74. Interference between ghost measurement beam 84 and the desired beams 72, 74 imposes a cyclic error term with a period of λ/2, where λ is the nominal wavelength of the light output by light source 60, on the electrical signal generated by sensor 70. In other words, when movable mirror 50 is displaced by a distance equal to λ/2, ghost measurement beam 84 imposes a full cycle of an approximately sinusoidal error on the electrical signal generated by sensor 70.
It is known in the art that the effect of ghost beams such as ghost reference beam 82 and ghost measurement beam 84 can be mitigated by tilting one or more of the surfaces of the interferometer at which light is reflected. Light reflected by such tilted reflective surface is neither parallel to nor orthogonal to the direction of incident light beam 62. Reflective surfaces that can be tilted include the reflective surface 42 of fixed mirror 40, the reflective surface 52 of movable mirror 50 and the beam-splitting surface 22 of PBS 20. The reflective surface of each mirror can be tilted such that the reflective surface is no longer orthogonal to the reflective surface of the other mirror. Beam-splitting surface 22 can be tilted such that it is oriented at an angle different from 45° to the normal to the reflective surface of each of the mirrors 40, 50. More than one of the above-mentioned reflective surfaces may be tilted.
Tilting one or more of the reflective surfaces of interferometer 10 mitigates the effect of ghost beams 82, 84 by causing the ghost beams to emerge from PBS 20 with their beam axes diverging from those of the desired beams, i.e., reference beam 72 and measurement beam 74. Consequently, at sensor 70, the beam axes of ghost beams 82, 84 diverge from those of desired beams 72, 74 and the beam axes of the ghost beams are spatially offset from those of the desired beams. The spatial offset of the beam axes of ghost beams 82, 84 from those of desired beams 72, 74 and the beam axes of ghost beams 82, 84 diverging from those of desired beams 72, 74 significantly reduce the level of the component of the electrical signal generated by sensor 70 in response to interference between ghost beams 82 and 84 on one hand and desired beams 72, 74 on the other hand.
However, tilting one or more of the reflective surfaces of interferometer 10 additionally introduces a spatial offset between the beam axes of reference beam 72 and measurement beam 74 as these beams emerge from PBS 20, even though the beam axes of reference beam 72 and measurement beam 74 typically remain parallel. The spatial offset between the beam axes of reference beam 72 and measurement beam 74 at sensor 70 undesirably reduces the amplitude of the desired interference signal, i.e., the component in the electrical signal generated by sensor 70 in response to interference between desired beams 72, 74. Thus, compared with an interferometer such as interferometer 10 having no tilted reflective surface, the desired interference component generated by an interferometer having at least one tilted reflective surface has a reduced amplitude and, hence, a reduced signal-to-noise ratio. This impairs the accuracy of the displacement measurements made using such interferometer.
What is needed, therefore, is an interferometer in which the effect of ghost beams is mitigated in a way does not degrade the signal-to-noise ratio of the desired interference component of the electrical signal generated by the sensor.